AAsymmetry nature in symmetry breakdown : Anti-PT-symmetry
Biswanath Rath
Physics Department , North Orissa University, Baripada -757003, Odisha, India, email : [email protected].
We discover present anti βPT- symmetry operator is a problematic non-hermitian operator. However the usage of similarity transformation we seriously change to a new non-hermitian operator underneath the identical commutating operator, Interestingly the use of new shape of PT- symmetry we reproduce preceding experimental outcomes and propose new nature of asymmetry in symmetry breakdown. Introduction
Since the development of PT- symmetry by Bender and Bottecher and subsequent. Development there have been developing hobby if symmetry breakdown. In order to provide an explanation of the symmetry breaking some creator used the matrix mannequin Hamiltonian as
π» = [π + ππ ππππ βπ + ππ] (1) Having eigenvalues πΒ±= ππ Β± βπ β π (2) At this point we would like to state that P-parity of the above mannequin Hamiltonian is π = [1 00 β1] (3) Hence the usage of property of T-time -reversal operator, π β1 ππ = βπ (4) We find {ππ, π»} β 0 (5) However the above complex operator posses a symmetry operator called C-symmetry whose explicit form is given as πΆ = [ π ππππ βπ] / βπ β π (6) It is trivial to find π Β±πΆ = Β±1 (7) Now we recommend a mannequin Hamiltonian besides the altering the nature of C as follows. 2. PT- symmetry operator under invariance C- symmetry Here we use similarity transformation π β1 π»π = β ππ (8) Where π = [ 0 1βπ 0] (9) Hence β ππ = [βπ + π βππ π + π] (10) The above operator can be written as β ππ = [βπ + π ππππ π + π] (11) Whose P-parity operator is πΆ ππ = βπ [βπ ππππ π ] (12) It is seen that both C and πΆ ππ carry the same eigenvalues. Further, P-parity of this operator is π = [β1 00 1] (13) It is easy to see that β = [π + π ππππ βπ + π] (14) also bear the same eigenvalue. Further [π», πΆ] = 0; [β , πΆ] = 0 (15) . Asymmetry nature in symmetry breaking in Here we consider different value of the papameter b and c and observe the nature of asymmetry.
Case-1, c=-3; a=8; b=0:10
In this case we reproduce the experimental results of Peng et.al (see fig- 3a) and Choi et.al (see fig-3a).
Figure 1 :
Asymmetry nature in symmetry breaking
Case-2 a=20;c=b;b=-10:0.
In this case we reproduce the experimental results of Choi et.al (see fig-3b)
Figure 2 :
Asymmetry nature in symmetry breaking
Case-3 a=8;c=b;b=-10:10.
In this case propose new nature of asymmetry.
Figure 3 :
Asymmetry nature on symmetry breaking
Case-4 a=20;c=b;b=-10:10 , here we plot eigenvalues square In this case propose new nature of asymmetry. Conclusion
In this paper we show that matrix is neither PT-symmetry nor anti- PT-symmetry. however the usage of similarity transformation some can locate a new form of PT-symmetry operator. Considering unique values of the parameter β a,b,cβ we find proto type of figs mirrored earlier. In addition to this we get different nature of asymmetry. Acknowledgement . Author is thankful to Smruti Rath for helping in English writing of the paper. Competing interest Author declares there is no competing interest.
Figure 4 :
Asymmetry nature in symmetry breaking eferences C. Bender and S .Boettcher, Real spectra in non-Hermitian Hamiltonians having PT-symmetry. Phys.Rev.Lett80(24),5243(1998). 2.
C. Bender , D .C .Brody and H. F. Jones, complex extension of quantum mechanics , Phys. Rev.ett 89(27_,270401(2002) and Erratum 92(11), 119902(2004). 3.
Peng.P.et.all,Ant-parity-time symmetry with flying atoms. Nature Phys. 12.,11391145(2016). 4.